Does an aura that was attached to a creature fall off when that creature gains protection from the aura's color?

Protection from a quality means the protected permanent cannot be enchanted by sources of that quality (among other things). This is the 'E' in the DEBT acronym: protection prevents Damage, Enchanting/Equipping, Blocking, and Targeting by the specified quality (CR 702.16b).

When an aura is already attached to a creature that then gains protection from the aura's color, the aura doesn't immediately fall off on its own — instead, the game checks state-based actions. State-based actions are checked continuously, and one of them (CR 704.5n) states: if an aura is attached to an object or player it cannot legally enchant, the aura is put into its owner's graveyard.

So the sequence is: (1) the creature gains protection, (2) the next time state-based actions are checked (before any player gets priority), (3) the aura is found to be illegally attached, and (4) it goes to its owner's graveyard. It does not simply 'fall off' neutrally — it is put into the graveyard, which can matter for triggered abilities.

Example: Your opponent's Grizzly Bears is enchanted by your Pacifism (a white aura). Your opponent casts Mother of Runes and activates her ability to give Grizzly Bears protection from white. Before anyone gets priority again, state-based actions are checked: Pacifism can no longer legally enchant Grizzly Bears, so it is put into your graveyard.

Note: if the aura has an ability that grants protection (like Flickering Ward), the aura won't make itself illegal in most cases because such auras are typically written carefully — but any external source granting protection will cause the aura to fall into the graveyard via state-based actions.